Precise positioning is a basic and important technology. Leading-edge technical fields including semiconductor manufacturing, liquid crystal manufacturing, and processing of optoelectronic devices require precise positioning apparatuses with high resolution.
Those including a combination of a linear motor and a static pressure guide are known as the precise positioning apparatuses. The static pressure guide supplies pressurized air, and floats a movable body to guide the movable body by using low viscosity of the air. Friction is hard to act on the guide. Accordingly, the static pressure guide is suitable for precise positioning. However, there are problems that the rigidity of the guide is low and the guide is sensitive to disturbances.
In recent years, instead of the static pressure guide, a rolling guide apparatus using a spherical or cylindrical rolling element has started being used. Reasons of this include an improvement in performance of a commercial rolling guide apparatus, and higher rigidity of the guide than the static pressure guide.
When a force is applied to a guide block of the rolling guide apparatus, the relationship between the applied force and a displacement caused generates hysteresis and exhibits non-linearity. The characteristic exhibiting non-linearity is called a non-linear spring characteristic, which is known to be present in a minute displacement region of several tens μm or below. The non-linear spring characteristic appears before the rolling element in the guide block starts rolling or when the rolling element reverses its rolling direction, and adversely affects the accuracy of precise positioning. For example, when arc motion is performed on a two-axis, X and Y, stage using the rolling guide apparatus, and an error between a target command position and an actual position is enlarged, a spike-like error (a quadrant glitch) occurs at quadrant changes. It is hard to eliminate the quadrant glitch even by using closed-loop control that feeds back the position of a table. Therefore, when the rolling guide apparatus is used, the handling of the non-linear spring characteristic is an important issue.
As how to handle the non-linear spring characteristic, Patent Literature 1 discloses an invention that, in a position control apparatus of a table that is guided by the rolling guide apparatus, obtains a friction force from the non-linear spring characteristic of a rolling guide apparatus on the basis of the velocity of the table and corrects the thrust of the table with the obtained friction force. In other words, the invention described in Patent Literature 1 assumes that the non-linear spring characteristic is one of friction characteristics and a friction force obtained from the non-linear spring characteristic is a disturbance of thrust.